Entrepreneurial Chess

Abstract

Although the combinatorial game EntrepreneurialChess (or Echess) was invented around 2005, this is our first publication devoted to it. A single Echess position begins with a Black king vs. a White king and a White rook on a quarter-infinite board, spanning the first quadrant of the xy-plane. In addition to the normal chess moves, Black is given the additional option of “cashing out”, which removes the board and converts the position into the integer \(x + y\), where [x, y] are the coordinates of his king’s position when he decides to cash out. Sums of Echess positions, played on different boards, span an unusually wide range of topics in combinatorial game theory. We find many interesting examples.

Keywords

Combinatorial games Chess

Mathematics Subject Classification

Notes

Acknowledgements

In 2005, Mark Pearson, then a student in Berlekamp’s graduate course on Combinatorial Game Theory, determined the means and temperatures shown in Figs. 15 and 26. All of our subsequent calculations have been done by hand. The authors wish to thank Eugenia Lee for creating the beautiful figures found throughout this paper. We are also indebted to David Berlekamp, both for his technical prowess in dealing with several mutually incompatible document preparation technologies, and for his help in debugging the mathematics in some of the sections of this paper. We are grateful for the valuable comments made by the referee and Editor. Finally, the authors wish to thank Bernhard von Stengel for his help in resolving some unforeseen typesetting problems as this paper went to press.